Question

The annual number of earthquakes registering at least 2.5 on the Richter Scale and having an epicenter within 40 miles of downtown Memphis follows a Poisson distribution with mean 6.5.

a. What is the probability that at least 9 such earthquakes will strike next year?

b. What is the probability that at least 25 such earthquakes will strike during the next 5 years?

Answer 1

Solution:

a.

We need to find the probability of Poisson distribution wil $\lambda = 6.5$

Remembering Poisson formula and using calculator:

$P_n(\xi=m) = \frac{\lambda^m}{m!}e^{-\lambda}$

$P_n(\xi\ge9)=1- P_n(\xi=0)-P_n(\xi=1)-...-P_n(\xi=8)=$

$=1 - 0.0015 - 0.00977-0.03176-0.06881-0.11182-0.14537-$

$-0.15748-0.14623-0.11882\approx0.20843$

b.

We can use the other Poisson distribution, for 5 years, where $\lambda=6.5*5=32.5$

Using inverted probability:

$P_n(\xi\ge25)= 1- P_n(\xi \le 24)= 1 - P_n(\xi=0) - P_n(\xi=1) - ...-P_n(\xi=24) =$

$= 1- 0.07536 =0.92464$

Answer:

a. $P_n(\xi\ge 9)=0.20843$

b.$P_n(\xi\ge 25)=0.92464$